Professor Uses Mathematics to Decode Beatles Tunes Article, Video

HALIFAX, Canada -- It is here, in a cluttered mathematician's office, under blackboards jammed with equations and functional analysis, that one of Western culture's greatest mysteries has finally been solved:

Why has no one been able to replicate the first chord in The Beatles' pop hit "A Hard Day's Night"?

Not stopping there, this sleuth is using math in his quest to answer an even more-elusive question, about the contested authorship of the Fab Four's "In My Life."

All You Need Is Math

See how Mr. Brown used math to figure out The Beatles' formula for success, listen to the clips he analyzed, and watch him perform his own Beatles-esque song.
"Whether they realize it or not, the best songwriters have always relied on mathematics," says Jason Brown, a mathematician who is tackling such puzzlers in between chairing the math department at Dalhousie University in Halifax, Nova Scotia, and playing his own Beatles-esque songs.

Since the 1960s, Beatles aficionados have pored over the group's recordings and memorabilia in search of answers to questions about both their music and their lives. An as-yet-unpublished Beatles track Paul McCartney recently mentioned made headlines internationally. Beatles historian Mark Lewisohn, who is working on a trilogy about the British rock group, examined church records to find the precise day that John Lennon met Mr. McCartney.

Math Professor Figures Formula for Beatles Success
Jason Brown listens to the Beatles with a uniquely analytical ear. The mathematics professor at Dalhousie University in Halifax, Nova Scotia, says he's figured out the math behind the best of the Fab Four. Now, using "mathematical tricks" he's picked up from the band, he's written a very Beatles-esque song of his own. WSJ's Christina Jeng reports.

Math Professor Jason Brown's 'A Million Whys'
Listen to Jason Brown's "A Million Whys." The mathematics professor at Dalhousie University in Halifax, Nova Scotia, used "mathematical tricks" he learned from analyzing songs by The Beatles and wrote a very Beatles-esque song of his own. WSJ's Christina Jeng reports.
Now Mr. Brown, 47 years old, is revisiting these questions from another angle. His approach is sparking controversy among fans who believe the band's mystique defies calculation. An article by Mr. Brown on his research published in Guitar Player magazine three years ago spawned heated discussions in both the math and music blogospheres.

"Some people thought what he was doing was sacrilegious," says Matt Blackett, an associate editor at Guitar Player. "As a fellow Beatles fanatic, I just thought it was awesome."

A spokesman for Mr. McCartney said he was unavailable and other former group members didn't respond to request for comment. Generally, the group has been evasive when faced with fans' efforts to dissect their work.

Growing up in the Toronto suburbs, Mr. Brown learned piano, but gave it up at age 12 for guitar, after hearing the Beatles' "Red Album," and becoming obsessed with the group. Like many Beatles fans, Mr. Brown was fascinated with the opening chord of "A Hard Day's Night." The chord has at least four sheet music variants, but nobody has ever quite replicated it, and the Beatles haven't revealed how they produced the complex sound. Mr. Brown said he spent hours experimenting before it occurred to him: "Music is basically just math."

It isn't surprising that Mr. Brown turned to mathematics. He talks about the lyrics of 1960s songwriter Randy Newman in terms of metamathematics. When he sees broccoli, he thinks of fractals, a concept in chaos theory. Piles of graph-theory tomes litter his office, and Greek letters and Roman numerals cover his chalkboard.

Mr. Brown realized he could use a discrete Fourier transform, a mathematical technique for breaking up complicated signals into simpler functions and known as DFT. He used digital equipment to show the chord as a series of numbers, tens of thousands per second, and then applied a DFT to convert the chord into dozens of simpler functions, each representing a single sound frequency.

Mr. Brown knew there is no such thing as a pure tone: Each instrument emits one sound for the note played and then sounds that are multiples of that note's frequency, as the string vibrates back on itself. Of his dozens of frequencies, some were background noise and some--the ones he wanted to ferret out--were the notes the Beatles struck.

The professor started making deductions. The loudest notes were likely Mr. McCartney's bass. The lowest had to be the original note played, since a string can generate waves along half or a third of its length, but not twice its length. But no matter how he divvied up the notes, something didn't fit.

It is well-documented that Mr. Harrison played a 12-string guitar for the recording of "A Hard Day's Night." For every guitar note played, there had to be another one octave higher, since his guitar strings were pressed down in pairs.

But three frequencies for an F note were left, none of which were an octave apart. Even if Mr. Brown assumed Mr. Lennon played one F note on his six-string guitar, Mr. Brown still had two unexplained frequencies.

After weeks of staring at six-decimal-place amplitude values, Mr. Brown suddenly remembered how, as a child, he used to stick his head inside his parents' grand piano to see how it worked. He ran to a nearby music shop, and poked his head inside the Yamahas there.

Sure enough, there were three strings under the F key, corresponding to the three sets of harmonics he had seen. Buried under the iconic guitar chord was a piano note.

Other problems have since yielded to Mr. Brown's mathematics. Fans have always marveled at Mr. Harrison's guitar solo in "A Hard Day's Night," a rapid-fire sequence of 1/16th notes, accompanied on piano, that seemed to require superhuman dexterity.

Mr. Brown noticed that a piano is strung differently in its lower octaves, with two strings, rather than three, under each hammer. He saw only two frequencies for each piano note in the guitar solo, suggesting that the solo had been played one octave lower than the recorded version sounded. It had also been played at half-speed, he concluded, then sped up on tape to make the released version sound as if had been played faster and at a higher octave.